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Ta b l e 7 . 4
Rearranged Table 7.3 separating the positive assessments from the negative
assessments.
Product
Attributes
Attributes
Total
Positive assessments Negative assessments
Atr 1 Atr 2 Atr 3 Atr 4 Atr 5 Atr 6 Atr 1 Atr 2 Atr 3 Atr 4 Atr 5 Atr 6
A
12
10
8
11
21
11
88
90
92
89
79
89
600
B
3 7 1 4 1 4 7 3 9 6 9 6 0
C
3 9 0 5 2 4 7 1 0 5 8 6 0
D
3 7 3 9 6 1 7 3 7 1 4 9 0
With the rearrangement shown in Table 7.4 the nonproportionate nature of the rows
for
A
and
C
is clear. Tables 7.3 and 7.4 both give the same CA analysis. The constant
row sums give weights that simplify the chi-squared distances. The precise nature of this
simplification is best seen by writing the Table 7.4 data matrix in algebraic form:
(
X
,
N
11
−
X
)
,
(7.46)
where
N
is the number of assessors. The row and column sums of (7.46), expressed
as diagonal matrices, are
R
∗
=
Nq
I
and
C
∗
=
(
C
,
Np
I
−
C
)
,where
C
is the usual
column-sum diagonal matrix of
X
itself. Note, that
R
has been eliminated from fur-
ther consideration and that
n
=
Npq
is the total sum of the extended data matrix. We
may now find formulae for the chi-squared distances using the results of Section 7.2.4
(see also Table 7.1). Thus, the row chi-squared distances are generated by the rows of
R
∗−
1
(
X
,
N
11
−
X
)
C
∗−
1
/
2
,
(7.47)
and the column chi-squared distances are generated by the columns of
R
∗−
1
/
2
(
X
,
N
11
−
X
)
C
∗−
1
.
(7.48)
From (7.47) the row chi-squared distances are generated by the rows of
(
Nq
)
−
1
(
X
,
−
X
)(
C
,
Np
I
−
C
)
−
1
/
2
(7.49)
because
11
C
∗−
1
/
2
represents a constant term added to every item of each column and
so has no effect on distance and may be eliminated. Furthermore, the same columns of
X
occur in both parts of (7.47); only the column weights change. Thus the
k
th attribute
contributes to the distance between products
i
and
i
the quantity
(
Nq
)
−
2
2
{
c
−
1
k
+
(
Np
−
c
k
)
−
1
(
x
ik
−
x
i
k
)
}
,
which simplifies to
Np
(
Nq
)
−
2
2
{
c
k
(
Np
−
c
k
)
}
−
1
(
x
ik
−
x
i
k
)
.
(7.50)
c
k
)
}
−
1
The factor
{
c
k
(
Np
−
c
k
)
}
is minimum (so
{
c
k
(
Np
−
has maximal weight in (7.50))
when
c
k
is close to zero or
Np
, and is maximum at
c
k
=
Np
/2. Greenacre (2007) refers
to it as a measure of polarization of the
k
th attribute.
